2-norm of a projector is greater than 1

Let [tex]P \in \textbf{C}^{m\times m}[/tex] be a projector. We prove that [tex]\left\| P \right\| _{2}\geq 1[/tex] , with equality if and only if P is an orthogonal projector.

I suppose we could use the formula [tex]\left\| P \right\| _{2}= max_{\left\| x \right\| _ {2} =1} \left\| Px \right\| _{2}[/tex] and use the fact that [tex]P^{2}=P[/tex] and [tex]P=P^{*}[/tex] (P* is the transpose conjugate of P).